Abstract
In recent years, various operators of fractional calculus (that is, calculus of integrals and derivatives of arbitrary real or complex orders) have been investigated and applied in many remarkably diverse fields of science and engineering. Many authors have demonstrated the usefulness of fractional calculus in the derivation of particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. The purpose of this paper is to present a certain class of the explicit particular solutions of the associated CauchyEuler fractional partial differential equation of arbitrary real or complex orders and their applications as follows:
where
MSC: 26A33, 33C10, 34A05.
Keywords:
fractional calculus; differential equation; partial differential equation; generalized Leibniz rule; analytic function; index law; linearity property; principle value; initial and boundary valueDedication
Dedicated to Professor Hari M Srivastava.
1 Introduction, definitions and preliminaries
The subject of fractional calculus (that is, derivatives and integrals of any real
or complex order) has gained importance and popularity during the past two decades
or so, due mainly to its demonstrated applications in numerous seemingly diverse fields
of science and engineering (cf.[115]). By applying the following definition of a fractional differential (that is, fractional derivative and fractional integral) of order
In this paper, we present a direct way to obtain explicit solutions of such types of the associated CauchyEuler fractional partial differential equation with initial and boundary values. The results are a coincidence that the solutions are obtained by the methods applying the Laplace transform with the residue theorem. In this paper, we present some useful definitions and preliminaries for the paper as follows.
If the function
and
where
and
then
First of all, we find it is worthwhile to recall here the following useful lemmas and properties associated with the fractional differintegration which is defined above.
Lemma 1.1 (Linearity property)
If the functions
for any constants
Lemma 1.2 (Index law)
If the function
Lemma 1.3 (Generalized Leibniz rule)
If the functions
where
Lemma 1.4 (Cauchy’s residue theorem)
Let Ω be a simple connected domain, and letCbe a simple closed positively oriented contour that lies in Ω. Iffis analytic insideCand on C, expect at the point
(I) Iffhas a simple pole at
(II) Iffhas a pole of orderkat
For a constanta,
Proof The proofs between ‘ν is not an integer’ and ‘ν is an integer’ are not coincident, so we mention the proof as follows.
In case of
for
In case of
Therefore we have Property 1.1 for arbitrary
Property 1.2For a constanta,
Property 1.3For a constanta,
The fractional derivative of a causal function
where
The Laplace transform of a function
wheresis the Laplace complex parameter. We recall from the fundamental formula (cf. [16])
2 Main results
Theorem 2.1The fractional partial differential equation
with
(a)
when the discriminant
(b)
when the discriminant
(c)
when the discriminant
Proof Suppose that
So that the given equation (2.1) becomes
Equation (2.4) leads to the auxiliary equation
That is,
and
are the two roots of the auxiliary equation (2.5). Thus,
There are three different cases to be considered, depending on whether the roots of
this quadratic equation (2.5) are distinct real roots, equal real roots (repeated
real roots), or complex roots (roots appear as a conjugate pair). The three cases
are due to the discriminant of the coefficients
• Case I: Distinct real roots (when
Let
where
• Case II: Repeated real roots (when
If the roots of Equation (2.5) are repeated, that is,
where
• Case III: Conjugate complex roots (when
If the roots of Equation (2.5) are the conjugate pair
where
In general,
forms a fundamental solution, where
Theorem 2.2The fractional partial differential equation
with
(a′)
when the discriminant
(b′)
when the discriminant
(c′)
when the discriminant
Proof The similarity between the forms of solutions of Equation (2.1) and solutions of a linear equation with constant coefficients of Equation (2.6) is not just a coincidence.
Suppose that
So that the given equation (2.6) becomes
If
then
and
The analysis of three cases is similar to Theorem 2.1, we can obtain each solution of the forms as follows:
□
Remark The constant λ in Equations (2.2) and (2.7) can be solved directly by constant initial value and constant boundary values (or by the numerical methods).
Corollary 2.1The fractional partial differential equation
with
(a″)
when the discriminant
(b″)
when the discriminant
(c″)
when the discriminant
Corollary 2.2The fractional partial differential equation
with
(
when the discriminant
(
when the discriminant
(
when the discriminant
3 Examples
Example 3.1 If the twodimensional harmonic equation
then it has solutions of the form
where
Solution Equation (3.1) is coincident to
We have the solution
by taking
Example 3.2 The fractional partial differential equation
Solution Putting
where the discriminant
If
The analysis of the case
Example 3.3 The fractional partial differential equation
Solution Putting
Then
That is,
Thus,
If the discriminant
□
Example 3.4 The fractional partial differential equation
Solution Putting
By the boundary condition
and the particular solution is
If we apply the Laplace transform to
The solution obtained by the method of Laplace transform and the residue theorem is a coincidence, which is our result above. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
SDL carried out the molecular genetic studies, participated in the sequence alignment and drafted the manuscript. CHL carried out the immunoassays. SMS participated in the sequence alignment.
Acknowledgements
The authors are deeply appreciative of the comments and suggestions offered by the referees for improving the quality and rigor of this paper. The present investigation was supported, in part, by the National Science Council of the Republic of China under Grant NSC1012115M033002.
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